Multiscale mathematical and computational models for biofluids and their applications to the study of the coupled eye-cerebral system

Multiscale coupling of nonlinear distributed and lumped fluid flow models is often necessary when modeling complex biological vascular systems. When interested in studying in details a specific segment of the vasculature, usually, to reduce simulations costs, a distributed partial differential equations (PDEs) model is used to simulate the segment of interest, while the rest of the vasculature is approximated using a lumped ordinary differential equations (ODEs) model. We propose a new splitting approach to numerically solve this multiscale problem in an efficient, accurate and affordable manner. The main novelty of the splitting scheme is that it ensures that the energy of the semi-discrete problem mirrors the behavior of the energy of the fully coupled problem. As a result, unconditional stability with respect to the time step choice is ensured without the need of sub-iterating between PDE and ODE sub-steps. We next illustrate the capabilities of this framework by applying it to the development of a multiscale model describing the coupled dynamics of different biofluids in the brain and in the eye.